The progressive modes give you full frame progressive capture, as opposed to interlaced. That is the different between a video look and a film look essentially.
24p the standard 3:2 pulldown used for telecine of film to video to make 24 frames per second into a 29.97 NTSC compatible signal while retaining the film look. 24p Advanced does the same thing but in a way that's much easier to reverse in editing- so you can edit without the pulldown at a true 24 frames per second.
You should be shooting whatever format you like but 24p advanced is this camera's strongest and most used mode.